Applications of the Statistical Model for Schrödinger Equations

Abstract

Based on the statistical mechanics for Schrödinger equations discussed in the preceding chapters, segregation of monkey populations was discussed in Nagasawa (1980) as a first trial:

“Assume there is a feeding place in a mountain (perhaps in Japan) where we find a population of monkeys (we assume that the movement of a monkey can be approximated as a sample path of a diffusion process, since he is hopping and jumping from one place to another). The feeding place attracts monkeys and most of them move toward the place and will eventually be distributed with the peak density at the centre (one-core distribution). Besides this, there is another equilibrium distribution, where monkeys come closer but do not reach the feeding place, and are distributed on a circle making a doughnut around the feeding place. When the one-core distribution is realized, the population is in the state of the lowest excitation, and in the doughnut distribution the population is an excited state. Moreover, it can be shown that if the population stays as a single group, no other equilibrium distribution exists, that is, in a state of higher excitation the population cannot keep itself as a single party but splits into at least two groups. Once it is divided into two parties, it can take one of three possible equilibrium distributions.” We call this segregation of a population. If two groups of monkeys are observed around the feeding place, we find three possible distributions, which are distinguished by different eigenvalues (= degrees of excitation of the population).1